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                    2.1 排序查找
            
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                    2.2 动态规划
            
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                    3.1 一致性模式与理论
            
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                    3.5 几种一致性方案
            
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                    4.1 线性规划
            
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                    4.2 整数规划
            
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                    4.3 目标规划
            
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                    4.4 非线性规划
            
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                    4.5 动态规划
            
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                    4.7 博弈论
            
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                    1 设计模式
            
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                    1.1 策略模式
            
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                    1.2 原型模式
            
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                    1.4 工厂模式
            
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                    1.5 桥接模式
            
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                    1.6 组合模式
            
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                    1.8 适配器模式
            
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                    1.9 装饰者模式
            
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                    1.10 享元模式
            
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                    1.11 代理模式
            
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                    1.12 中间件模式
            
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                    2 网络编程
            
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                    2.1 长连接
            
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                    2.2 MQ
            
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                    3 语言应用
            
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                    3.1.1 原理及构成
            
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                    3.1.2 异步机制
            
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                    3.1.3 一些核心概念
            
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                    3.1.4 web应用: koa框架
            
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                    3.1.5 koa源码分析
            
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                    3.1.6 libuv网络库应用
            
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                    3.1.7 性能调优
            
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                    3.1.8 其他: Node爬虫
            
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                    3.1.9 其他: Node编写Mapreduce
            
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                    3.2 C/C++
            
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                    3.2.1 面向对象
            
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                    3.2.2 进程通信
            
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                    3.2.3 线程管理
            
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                    3.2.4 内存管理
            
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                    3.2.5 网络编程
            
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                    3.2.6 构建工具
            
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        <li class="chapter " data-level="1.3.3.2.7" data-path="../program/c/thread_pool.html">
            
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                    3.2.7 其他: 线程池
            
                </a>
            

            
        </li>
    

            </ul>
            
        </li>
    

            </ul>
            
        </li>
    

            </ul>
            
        </li>
    
        <li class="chapter " data-level="1.4" data-path="../engineering/ci_deplopment/">
            
                <a href="../engineering/ci_deplopment/">
            
                    
                    三 搭建部署
            
                </a>
            

            
            <ul class="articles">
                
    
        <li class="chapter " data-level="1.4.1" data-path="../engineering/ci_deplopment/test.html">
            
                <a href="../engineering/ci_deplopment/test.html">
            
                    
                    3.1 测试
            
                </a>
            

            
            <ul class="articles">
                
    
        <li class="chapter " data-level="1.4.1.1" data-path="../engineering/ci_deplopment/test/unit_testing.html">
            
                <a href="../engineering/ci_deplopment/test/unit_testing.html">
            
                    
                    3.1.1 单元测试
            
                </a>
            

            
        </li>
    
        <li class="chapter " data-level="1.4.1.2" data-path="../engineering/ci_deplopment/test/stress_testing.html">
            
                <a href="../engineering/ci_deplopment/test/stress_testing.html">
            
                    
                    3.1.2 压力测试
            
                </a>
            

            
        </li>
    

            </ul>
            
        </li>
    
        <li class="chapter " data-level="1.4.2" data-path="../engineering/ci_deplopment/ci_readme.html">
            
                <a href="../engineering/ci_deplopment/ci_readme.html">
            
                    
                    3.2 集成
            
                </a>
            

            
            <ul class="articles">
                
    
        <li class="chapter " data-level="1.4.2.1" data-path="../engineering/ci_deplopment/ci/gitlab_ci.html">
            
                <a href="../engineering/ci_deplopment/ci/gitlab_ci.html">
            
                    
                    3.2.1 GitLab CI
            
                </a>
            

            
        </li>
    
        <li class="chapter " data-level="1.4.2.2" data-path="../engineering/ci_deplopment/ci/spinnaker.html">
            
                <a href="../engineering/ci_deplopment/ci/spinnaker.html">
            
                    
                    3.2.2 Spinnaker
            
                </a>
            

            
        </li>
    

            </ul>
            
        </li>
    
        <li class="chapter " data-level="1.4.3" data-path="../engineering/ci_deplopment/build.html">
            
                <a href="../engineering/ci_deplopment/build.html">
            
                    
                    3.3 部署
            
                </a>
            

            
            <ul class="articles">
                
    
        <li class="chapter " data-level="1.4.3.1" data-path="../engineering/ci_deplopment/k8s/k8s.html">
            
                <a href="../engineering/ci_deplopment/k8s/k8s.html">
            
                    
                    3.3.1 k8s
            
                </a>
            

            
        </li>
    
        <li class="chapter " data-level="1.4.3.2" data-path="../engineering/ci_deplopment/rancher/rancher.html">
            
                <a href="../engineering/ci_deplopment/rancher/rancher.html">
            
                    
                    3.3.2 rancher
            
                </a>
            

            
        </li>
    
        <li class="chapter " data-level="1.4.3.3" data-path="../engineering/ci_deplopment/swarm/swarm.html">
            
                <a href="../engineering/ci_deplopment/swarm/swarm.html">
            
                    
                    3.3.3 swarm
            
                </a>
            

            
        </li>
    

            </ul>
            
        </li>
    
        <li class="chapter " data-level="1.4.4" data-path="../engineering/ci_deplopment/maintenance.html">
            
                <a href="../engineering/ci_deplopment/maintenance.html">
            
                    
                    3.4 监控
            
                </a>
            

            
            <ul class="articles">
                
    
        <li class="chapter " data-level="1.4.4.1" data-path="../engineering/ci_deplopment/monitor.html">
            
                <a href="../engineering/ci_deplopment/monitor.html">
            
                    
                    3.4.1 容器监控
            
                </a>
            

            
        </li>
    
        <li class="chapter " data-level="1.4.4.2" data-path="../engineering/ci_deplopment/trace.html">
            
                <a href="../engineering/ci_deplopment/trace.html">
            
                    
                    3.4.2 系统监控
            
                </a>
            

            
        </li>
    
        <li class="chapter " data-level="1.4.4.3" data-path="../engineering/ci_deplopment/app1.md">
            
                <span>
            
                    
                    3.4.2 应用监控
            
                </a>
            

            
        </li>
    

            </ul>
            
        </li>
    
        <li class="chapter " data-level="1.4.5" data-path="../engineering/ha/ha.html">
            
                <a href="../engineering/ha/ha.html">
            
                    
                    3.5 高可用
            
                </a>
            

            
            <ul class="articles">
                
    
        <li class="chapter " data-level="1.4.5.1" data-path="../engineering/ha/client.html">
            
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                    3.5.1 接入层高可用
            
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        <li class="chapter " data-level="1.4.5.2" data-path="../engineering/ha/service.html">
            
                <a href="../engineering/ha/service.html">
            
                    
                    3.5.2 服务层高可用
            
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        <li class="chapter " data-level="1.4.5.3" data-path="../engineering/ha/share.html">
            
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                    3.5.3 存储层高可用
            
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        <li class="chapter " data-level="1.4.6" data-path="../engineering/ci_deplopment/hc/hc.html">
            
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                    3.6 高并发
            
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        <li class="chapter " data-level="1.4.6.1" data-path="../engineering/ci_deplopment/hc/dns.html">
            
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                    3.6.1 基于DNS的负载均衡
            
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        <li class="chapter " data-level="1.4.6.2" data-path="../engineering/ci_deplopment/hc/lvs.html">
            
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                    3.6.2 基于Lvs的负载均衡
            
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        <li class="chapter " data-level="1.4.6.3" data-path="../engineering/ci_deplopment/hc/haproxy.html">
            
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                    3.6.3 基于haproxy的负载均衡
            
                </a>
            

            
        </li>
    

            </ul>
            
        </li>
    

            </ul>
            
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        <li class="chapter " data-level="1.5" >
            
                <span>
            
                    
                    四 架构模式
            
                </span>
            

            
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        <li class="chapter " data-level="1.5.1" >
            
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                    1 微服务架构设计与实践
            
                </span>
            

            
            <ul class="articles">
                
    
        <li class="chapter " data-level="1.5.1.1" data-path="../program/java/spring-cloud.html">
            
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                    1.1 微服务概况
            
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        </li>
    
        <li class="chapter " data-level="1.5.1.2" data-path="../program/java/spring-eureka.html">
            
                <a href="../program/java/spring-eureka.html">
            
                    
                    1.2 服务注册及发现
            
                </a>
            

            
        </li>
    
        <li class="chapter " data-level="1.5.1.3" data-path="../program/java/spring-feign.html">
            
                <a href="../program/java/spring-feign.html">
            
                    
                    1.3 服务调用
            
                </a>
            

            
        </li>
    
        <li class="chapter " data-level="1.5.1.4" data-path="../program/java/spring-zuul.html">
            
                <a href="../program/java/spring-zuul.html">
            
                    
                    1.4 网关及负载均衡
            
                </a>
            

            
        </li>
    
        <li class="chapter " data-level="1.5.1.5" data-path="../program/java/spring-hystrix.html">
            
                <a href="../program/java/spring-hystrix.html">
            
                    
                    1.5 熔断、限流、扩缩容
            
                </a>
            

            
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        <li class="chapter " data-level="1.5.1.6" data-path="../program/java/spring-config.html">
            
                <a href="../program/java/spring-config.html">
            
                    
                    1.6 云配置中心
            
                </a>
            

            
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        <li class="chapter " data-level="1.5.1.7" data-path="../program/java/zipkin-tracing.html">
            
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                    1.7 分布式链路追踪
            
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        <li class="chapter " data-level="1.5.1.8" data-path="../program/java/swagger.html">
            
                <a href="../program/java/swagger.html">
            
                    
                    1.8 接口文档管理
            
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                    2 领域驱动模型
            
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        <li class="chapter " data-level="1.5.2.1" data-path="../domain/domain.html">
            
                <a href="../domain/domain.html">
            
                    
                    2.1 领域--Domain
            
                </a>
            

            
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        <li class="chapter " data-level="1.5.2.2" data-path="../domain/entity.md">
            
                <span>
            
                    
                    2.2 领域实体--Entity
            
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        <li class="chapter " data-level="1.5.2.3" data-path="../domain/object.html">
            
                <a href="../domain/object.html">
            
                    
                    2.3 值对象--Value Object
            
                </a>
            

            
        </li>
    
        <li class="chapter " data-level="1.5.2.4" data-path="../domain/aggregate.html">
            
                <a href="../domain/aggregate.html">
            
                    
                    2.4 聚合--Aggregate
            
                </a>
            

            
        </li>
    
        <li class="chapter " data-level="1.5.2.5" data-path="../domain/service.html">
            
                <a href="../domain/service.html">
            
                    
                    2.5 服务--Services
            
                </a>
            

            
        </li>
    
        <li class="chapter " data-level="1.5.2.6" data-path="../domain/event.html">
            
                <a href="../domain/event.html">
            
                    
                    2.6 领域事件--Event
            
                </a>
            

            
        </li>
    
        <li class="chapter " data-level="1.5.2.7" data-path="../domain/framework.html">
            
                <a href="../domain/framework.html">
            
                    
                    2.7 几种常见的DDD架构
            
                </a>
            

            
        </li>
    

            </ul>
            
        </li>
    
        <li class="chapter " data-level="1.5.3" >
            
                <span>
            
                    
                    3 架构技术的演化与趋势
            
                </span>
            

            
            <ul class="articles">
                
    
        <li class="chapter " data-level="1.5.3.1" >
            
                <span>
            
                    
                    3.1 架构技术的发展历史及现状
            
                </span>
            

            
        </li>
    
        <li class="chapter " data-level="1.5.3.2" >
            
                <span>
            
                    
                    3.2 Serverless无服务架构概念
            
                </span>
            

            
        </li>
    
        <li class="chapter " data-level="1.5.3.3" >
            
                <span>
            
                    
                    3.3 ServerMesh网格服务概念
            
                </span>
            

            
        </li>
    

            </ul>
            
        </li>
    

            </ul>
            
        </li>
    
        <li class="chapter " data-level="1.6" >
            
                <span>
            
                    
                    五 其他
            
                </span>
            

            
            <ul class="articles">
                
    
        <li class="chapter " data-level="1.6.1" data-path="../story/talk.html">
            
                <a href="../story/talk.html">
            
                    
                    5.1 小伙子,放松!
            
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            </ul>
            
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                                <h3 id="&#x4E00;&#x3001;&#x7EBF;&#x6027;&#x89C4;&#x5212;">&#x4E00;&#x3001;&#x7EBF;&#x6027;&#x89C4;&#x5212;</h3>
<p>&#xA0;&#xA0;&#xA0;&#xA0;&#x7EBF;&#x6027;&#x89C4;&#x5212;&#x662F;&#x8FD0;&#x7B79;&#x5B66;&#x7684;&#x91CD;&#x8981;&#x7EC4;&#x6210;&#x3001;&#x73B0;&#x4EE3;&#x8BA1;&#x7B97;&#x673A;&#x7684;&#x9AD8;&#x901F;&#x53D1;&#x5C55;&#x4F7F;&#x5176;&#x5728;&#x7ECF;&#x6D4E;&#x3001;&#x7269;&#x6D41;&#x3001;&#x51B3;&#x7B56;&#x5206;&#x6790;&#x3001;&#x519B;&#x4E8B;&#x7B49;&#x65B9;&#x9762;&#x53D1;&#x6325;&#x8D8A;&#x6765;&#x8D8A;&#x91CD;&#x8981;&#x7684;&#x4F5C;&#x7528;&#xFF0C;&#x5176;&#x4E3B;&#x8981;&#x7528;&#x4E8E;&#x5408;&#x7406;&#x5206;&#x914D;&#x8D44;&#x6E90;&#x3001;&#x4EE5;&#x5982;&#x4F55;&#x7EC4;&#x7EC7;&#x751F;&#x4EA7;&#x7BA1;&#x7406;&#xFF0C;&#x901A;&#x8FC7;&#x5EFA;&#x7ACB;&#x6570;&#x5B66;&#x6A21;&#x578B;&#x6765;&#x8C03;&#x6574;&#x5404;&#x9879;&#x6307;&#x6807;&#x8FBE;&#x5230;&#x6700;&#x4F18;&#x5316;&#x76EE;&#x6807;&#xFF0C;&#x83B7;&#x5F97;&#x7ECF;&#x6D4E;&#x6548;&#x76CA;&#x6700;&#x5927;&#x5316;</p>
<ul>
<li><strong>1.1 &#x7EBF;&#x6027;&#x89C4;&#x5212;&#x57FA;&#x672C;&#x5B9A;&#x4E49;</strong></li>
</ul>
<pre><code>1&#x3001;&#x51B3;&#x7B56;&#x53D8;&#x91CF;: &#x5982;&#x95EE;&#x9898;&#x4E2D;&#x7684;&#x53D8;&#x91CF;&#x7EC4;(x1&#x3001;x2&#x3001;x3... xn)&#x7B49;&#x4EE3;&#x8868;&#x4E86;&#x4E00;&#x79CD;&#x51B3;&#x7B56;&#x65B9;&#x6848;&#xFF0C;&#x5176;&#x4E2D;&#x7684;&#x51B3;&#x7B56;&#x53D8;&#x91CF;&#x53D6;&#x503C;&#x975E;&#x8D1F;&#x65E2;xj &gt;= 0 (j=1&#x3001;2&#x3001;3...n)
2&#x3001;&#x7EA6;&#x675F;&#x6761;&#x4EF6;: &#x51B3;&#x7B56;&#x53D8;&#x91CF;&#x9700;&#x8981;&#x6EE1;&#x8DB3;&#x7684;&#x5982;&#x4E0A;&#x8FF0;&#x5404;&#x9879;&#x4E0D;&#x7B49;&#x5F0F;&#x6216;&#x7B49;&#x5F0F;
3&#x3001;&#x76EE;&#x6807;&#x51FD;&#x6570;: &#x5173;&#x4E8E;&#x51B3;&#x7B56;&#x53D8;&#x91CF;&#x7684;&#x7EBF;&#x6027;&#x51FD;&#x6570;&#xFF0C;&#x8FD9;&#x4E2A;&#x76EE;&#x6807;&#x51FD;&#x6570;&#x9700;&#x8981;&#x6EE1;&#x8DB3;&#x5728;&#x7EA6;&#x675F;&#x6761;&#x4EF6;&#x4E0B;&#x5B9E;&#x73B0;&#x6700;&#x5927;&#x6216;&#x6700;&#x5C0F;&#x5316;
</code></pre><ul>
<li><strong>1.2 &#x7EBF;&#x6027;&#x89C4;&#x5212;&#x6570;&#x5B66;&#x6A21;&#x578B;&#x53CA;&#x6807;&#x51C6;&#x5F62;&#x5F0F;</strong></li>
</ul>
<p>&#xA0;&#xA0;&#xA0;&#xA0;&#x7EBF;&#x6027;&#x95EE;&#x9898;&#x6C42;&#x89E3;&#x901A;&#x5E38;&#x6709;&#x56FE;&#x89E3;&#x6CD5;&#x548C;&#x6807;&#x51C6;&#x5F62;&#x4E24;&#x79CD;&#xFF0C;&#x56FE;&#x89E3;&#x6CD5;&#x7684;&#x65B9;&#x5F0F;&#x591A;&#x7528;&#x4E8E;&#x4E8C;&#x5143;&#x7EA6;&#x675F;&#x6761;&#x4EF6;&#xFF0C;&#x6807;&#x51C6;&#x5F62;&#x5F0F;&#x5219;&#x662F;&#x4E00;&#x79CD;&#x901A;&#x7528;&#x6C42;&#x89E3;&#x6A21;&#x5F0F;&#xFF0C;&#x56FE;&#x89E3;&#x65B9;&#x5F0F;&#x76F8;&#x5BF9;&#x6BD4;&#x8F83;&#x7B80;&#x5355;&#xFF0C;&#x8FD9;&#x91CC;&#x53EA;&#x4ECB;&#x7ECD;&#x6807;&#x51C6;&#x6A21;&#x5F0F;</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>m</mi><mi>a</mi><mi>x</mi><mi>Z</mi><mo>=</mo><msubsup><mo>&#x2211;</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>c</mi><mi>j</mi></msub><msub><mi>x</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">
maxZ = \sum_{j=1}^n c_jx_j
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.6513970000000007em;"></span><span class="strut bottom" style="height:3.0651740000000007em;vertical-align:-1.4137769999999998em;"></span><span class="base displaystyle textstyle uncramped"><span class="mord mathit">m</span><span class="mord mathit">a</span><span class="mord mathit">x</span><span class="mord mathit" style="margin-right:0.07153em;">Z</span><span class="mrel">=</span><span class="mop op-limits"><span class="vlist"><span style="top:1.1776689999999999em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mathrm mtight">1</span></span></span></span><span style="top:-0.000005000000000254801em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span><span style="top:-1.2500050000000005em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle uncramped mtight"><span class="mord mathit mtight">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span><span class="mord"><span class="mord mathit">c</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span></span></span></span></span> </p>
<center>
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><mo fence="true">{</mo><mtable><mtr><mtd><mrow><msub><mi>a</mi><mrow><mn>1</mn><mn>1</mn></mrow></msub><msub><mi>x</mi><mrow><mn>1</mn><mn>1</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><msub><mi>a</mi><mrow><mn>1</mn><mn>2</mn></mrow></msub><msub><mi>x</mi><mrow><mn>1</mn><mn>2</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><msub><mi>a</mi><mrow><mn>1</mn><mn>3</mn></mrow></msub><msub><mi>x</mi><mrow><mn>1</mn><mn>3</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo>+</mo><mtext>&#xA0;</mtext><msub><mi>a</mi><mrow><mn>1</mn><mn>2</mn></mrow></msub><msub><mi>x</mi><mrow><mn>1</mn><mn>2</mn></mrow></msub><mtext>&#xA0;</mtext><mo>=</mo><mtext>&#xA0;</mtext><msub><mi>b</mi><mn>1</mn></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>a</mi><mrow><mn>2</mn><mn>1</mn></mrow></msub><msub><mi>x</mi><mrow><mn>2</mn><mn>1</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><msub><mi>a</mi><mrow><mn>2</mn><mn>2</mn></mrow></msub><msub><mi>x</mi><mrow><mn>2</mn><mn>2</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><msub><mi>a</mi><mrow><mn>2</mn><mn>3</mn></mrow></msub><msub><mi>x</mi><mrow><mn>2</mn><mn>3</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo>+</mo><mtext>&#xA0;</mtext><msub><mi>a</mi><mrow><mn>2</mn><mn>4</mn></mrow></msub><msub><mi>x</mi><mrow><mn>2</mn><mn>4</mn></mrow></msub><mtext>&#xA0;</mtext><mo>=</mo><mtext>&#xA0;</mtext><msub><mi>b</mi><mn>2</mn></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>a</mi><mrow><mn>3</mn><mn>1</mn></mrow></msub><msub><mi>x</mi><mrow><mn>3</mn><mn>1</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><msub><mi>a</mi><mrow><mn>3</mn><mn>2</mn></mrow></msub><msub><mi>x</mi><mrow><mn>3</mn><mn>2</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><msub><mi>a</mi><mrow><mn>3</mn><mn>3</mn></mrow></msub><msub><mi>x</mi><mrow><mn>3</mn><mn>3</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo>+</mo><mtext>&#xA0;</mtext><msub><mi>a</mi><mrow><mn>3</mn><mn>4</mn></mrow></msub><msub><mi>x</mi><mrow><mn>3</mn><mn>4</mn></mrow></msub><mtext>&#xA0;</mtext><mo>=</mo><mtext>&#xA0;</mtext><msub><mi>b</mi><mn>3</mn></msub></mrow></mtd></mtr><mtr><mtd><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>a</mi><mrow><mi>m</mi><mn>1</mn></mrow></msub><msub><mi>x</mi><mrow><mi>m</mi><mn>1</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><msub><mi>a</mi><mrow><mi>m</mi><mn>2</mn></mrow></msub><msub><mi>x</mi><mrow><mi>m</mi><mn>2</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><msub><mi>a</mi><mrow><mi>m</mi><mn>3</mn></mrow></msub><msub><mi>x</mi><mrow><mi>m</mi><mn>3</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo>+</mo><msub><mi>a</mi><mrow><mi>m</mi><mi>n</mi></mrow></msub><msub><mi>x</mi><mrow><mi>m</mi><mi>n</mi></mrow></msub><mtext>&#xA0;</mtext><mo>=</mo><msub><mi>b</mi><mi>m</mi></msub><mtext>&#xA0;</mtext></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><mtext>&#xA0;</mtext><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mtext>&#xA0;</mtext><msub><mi>x</mi><mn>3</mn></msub><mo separator="true">,</mo><mtext>&#xA0;</mtext><msub><mi>x</mi><mn>4</mn></msub><mtext>&#xA0;</mtext><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><msub><mi>x</mi><mi>n</mi></msub><mtext>&#xA0;</mtext><mo>&#x2265;</mo><mn>0</mn><mtext>&#xA0;</mtext></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \left\{
\begin{aligned}
a_{11}x_{11}\ +\  a_{12}x_{12}\ +\ a_{13}x_{13}\ +\ ...  +\ a_{12}x_{12}\ =\ b_1 \\
a_{21}x_{21}\ +\  a_{22}x_{22}\ +\ a_{23}x_{23}\ +\ ...  +\ a_{24}x_{24}\ =\ b_2 \\
a_{31}x_{31}\ +\  a_{32}x_{32}\ +\ a_{33}x_{33}\ +\ ...  +\ a_{34}x_{34}\ =\ b_3 \\
... \\
a_{m1}x_{m1}\ +  a_{m2}x_{m2}\ + a_{m3}x_{m3}\ + ...  + a_{mn}x_{mn}\ = b_m\ \\
x_1,\ x_2,\ x_3,\ x_4\ ,...x_n\  \geq 0\ \\
\end{aligned}
\right.
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:4.4500199999999985em;"></span><span class="strut bottom" style="height:8.400039999999999em;vertical-align:-3.9500200000000003em;"></span><span class="base textstyle uncramped"><span class="minner textstyle uncramped"><span class="mopen style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing mult"><span class="vlist"><span style="top:3.0500100000000003em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23A9;</span></span></span><span style="top:3.0500100000000003em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:2.7500100000000005em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:2.4500100000000007em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:2.150010000000001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:1.8500100000000008em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:1.5500100000000008em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:1.2500100000000007em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:0.9500100000000007em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:-0.000009999999999399378em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23A8;</span></span></span><span style="top:-1.1500099999999993em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:-1.4500099999999994em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:-1.7500099999999994em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:-2.0500099999999994em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:-2.3500099999999993em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:-2.650009999999999em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:-2.950009999999999em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:-3.2500099999999987em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:-3.5500199999999986em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23A7;</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist"><span style="top:-3.6100000000000003em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="mord textstyle uncramped"><span class="mord"><span class="mord mathit">a</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span><span class="mord mathrm mtight">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span><span class="mord mathrm mtight">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord"><span class="mspace">&#xA0;</span><span class="mord mathit">a</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span><span class="mord mathrm mtight">2</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span><span class="mord mathrm mtight">2</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord"><span class="mspace">&#xA0;</span><span class="mord mathit">a</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span><span class="mord mathrm mtight">3</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span><span class="mord mathrm mtight">3</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">.</span></span><span class="mord mathrm">.</span><span class="mord mathrm">.</span><span class="mbin">+</span><span class="mord"><span class="mspace">&#xA0;</span><span class="mord mathit">a</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span><span class="mord mathrm mtight">2</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span><span class="mord mathrm mtight">2</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mrel"><span class="mspace">&#xA0;</span><span class="mrel">=</span></span><span class="mord"><span class="mspace">&#xA0;</span><span class="mord mathit">b</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span></span></span><span style="top:-2.4100000000000006em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="mord textstyle uncramped"><span class="mord"><span class="mord mathit">a</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span><span class="mord mathrm mtight">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span><span class="mord mathrm mtight">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord"><span class="mspace">&#xA0;</span><span class="mord mathit">a</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span><span class="mord mathrm mtight">2</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span><span class="mord mathrm mtight">2</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord"><span class="mspace">&#xA0;</span><span class="mord mathit">a</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span><span class="mord mathrm mtight">3</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span><span class="mord mathrm mtight">3</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">.</span></span><span class="mord mathrm">.</span><span class="mord mathrm">.</span><span class="mbin">+</span><span class="mord"><span class="mspace">&#xA0;</span><span class="mord mathit">a</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span><span class="mord mathrm mtight">4</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span><span class="mord mathrm mtight">4</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mrel"><span class="mspace">&#xA0;</span><span class="mrel">=</span></span><span class="mord"><span class="mspace">&#xA0;</span><span class="mord mathit">b</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span></span></span><span style="top:-1.2100000000000006em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="mord textstyle uncramped"><span class="mord"><span class="mord mathit">a</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">3</span><span class="mord mathrm mtight">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">3</span><span class="mord mathrm mtight">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord"><span class="mspace">&#xA0;</span><span class="mord mathit">a</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">3</span><span class="mord mathrm mtight">2</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">3</span><span class="mord mathrm mtight">2</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord"><span class="mspace">&#xA0;</span><span class="mord mathit">a</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">3</span><span class="mord mathrm mtight">3</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">3</span><span class="mord mathrm mtight">3</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">.</span></span><span class="mord mathrm">.</span><span class="mord mathrm">.</span><span class="mbin">+</span><span class="mord"><span class="mspace">&#xA0;</span><span class="mord mathit">a</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">3</span><span class="mord mathrm mtight">4</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">3</span><span class="mord mathrm mtight">4</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mrel"><span class="mspace">&#xA0;</span><span class="mrel">=</span></span><span class="mord"><span class="mspace">&#xA0;</span><span class="mord mathit">b</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span></span></span><span style="top:-0.010000000000000397em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="mord textstyle uncramped"><span class="mord mathrm">.</span><span class="mord mathrm">.</span><span class="mord mathrm">.</span></span></span><span style="top:1.1899999999999997em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="mord textstyle uncramped"><span class="mord"><span class="mord mathit">a</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">m</span><span class="mord mathrm mtight">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">m</span><span class="mord mathrm mtight">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord"><span class="mord mathit">a</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">m</span><span class="mord mathrm mtight">2</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">m</span><span class="mord mathrm mtight">2</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord"><span class="mord mathit">a</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">m</span><span class="mord mathrm mtight">3</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">m</span><span class="mord mathrm mtight">3</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord mathrm">.</span><span class="mord mathrm">.</span><span class="mord mathrm">.</span><span class="mbin">+</span><span class="mord"><span class="mord mathit">a</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">m</span><span class="mord mathit mtight">n</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">m</span><span class="mord mathit mtight">n</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mrel"><span class="mspace">&#xA0;</span><span class="mrel">=</span></span><span class="mord"><span class="mord mathit">b</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight">m</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mspace">&#xA0;</span></span></span><span style="top:2.39em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="mord textstyle uncramped"><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mpunct">,</span><span class="mord"><span class="mspace">&#xA0;</span><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mpunct">,</span><span class="mord"><span class="mspace">&#xA0;</span><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mpunct">,</span><span class="mord"><span class="mspace">&#xA0;</span><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">4</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mpunct"><span class="mspace">&#xA0;</span><span class="mpunct">,</span></span><span class="mord mathrm">.</span><span class="mord mathrm">.</span><span class="mord mathrm">.</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mrel"><span class="mspace">&#xA0;</span><span class="mrel">&#x2265;</span></span><span class="mord mathrm">0</span><span class="mspace">&#xA0;</span></span></span><span style="top:3.5900000000000003em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="mord textstyle uncramped"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span></span><span class="mclose sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span>
</center>

<pre><code>&#x6807;&#x51C6;&#x6A21;&#x578B;:
1&#x3001;&#x76EE;&#x6807;&#x51FD;&#x6570;&#x6700;&#x5927;&#x5316;: maxZ&#xFF0C;&#x82E5;&#x662F;minZ&#x5F62;&#x5F0F;&#xFF0C;&#x9700;&#x8981;&#x5C06;&#x5176;&#x8F6C;&#x4E3A;max&#x5F62;&#x5F0F;&#xFF0C;&#x6B64;&#x65F6;&#x6700;&#x4F18;&#x89E3;&#x4FDD;&#x6301;&#x4E0D;&#x53D8;
2&#x3001;&#x677E;&#x5F1B;&#x53D8;&#x91CF;&#x4E0E;&#x5269;&#x4F59;&#x53D8;&#x91CF;: &#x7EA6;&#x675F;&#x6761;&#x4EF6;&#x662F;&#x5C0F;&#x4E8E;&#x7B49;&#x4E8E;&#x578B;&#xFF0C;&#x5728;&#x4E0D;&#x7B49;&#x5F0F;&#x5DE6;&#x8FB9;&#x6DFB;&#x52A0;&#x65B0;&#x53D8;&#x91CF;--&#x677E;&#x5F1B;&#x53D8;&#x91CF;&#x4F7F;&#x5176;&#x4E3A;&#x7B49;&#x5F0F;, &#x82E5;&#x4E0D;&#x7B49;&#x5F0F;&#x662F;&#x5927;&#x4E8E;&#x7B49;&#x4E8E;&#x578B;&#xFF0C;&#x5728;&#x4E0D;&#x7B49;&#x5F0F;&#x5DE6;&#x8FB9;&#x51CF;&#x53BB;&#x4E00;&#x4E2A;&#x53D8;&#x91CF;--&#x5269;&#x4F59;&#x53D8;&#x91CF;&#xFF0C;&#x5C06;&#x4E0D;&#x7B49;&#x5F0F;&#x6539;&#x4E3A;&#x7B49;&#x5F0F;
3&#x3001;&#x7EA6;&#x675F;&#x53D8;&#x91CF;&#x4E0E;&#x4E0D;&#x7B49;&#x5F0F;&#x53F3;&#x7AEF;&#x9879;&#x975E;&#x8D1F;: &#x82E5;&#x4E0D;&#x7B49;&#x5F0F;&#x53F3;&#x7AEF;&#x9879;bj&#x4E3A;&#x8D1F;, &#x5C06;&#x4E0D;&#x7B49;&#x5F0F;&#x6539;&#x53D8;&#x65B9;&#x5411;
</code></pre><hr>
<h3 id="&#x4E8C;&#x3001;&#x7EBF;&#x6027;&#x89C4;&#x5212;&#x95EE;&#x9898;&#x6C42;&#x89E3;">&#x4E8C;&#x3001;&#x7EBF;&#x6027;&#x89C4;&#x5212;&#x95EE;&#x9898;&#x6C42;&#x89E3;</h3>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>m</mi><mi>a</mi><mi>x</mi><mi>Z</mi><mo>=</mo><mn>2</mn><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mn>3</mn><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mn>3</mn><mi mathvariant="normal">.</mi><mn>2</mn><mn>4</mn><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><mn>0</mn><mi mathvariant="normal">.</mi><mn>5</mn><msub><mi>x</mi><mn>4</mn></msub><mo>+</mo><mn>1</mn><mi mathvariant="normal">.</mi><mn>2</mn><msub><mi>x</mi><mn>5</mn></msub><mo>+</mo><mn>2</mn><msub><mi>x</mi><mn>6</mn></msub></mrow><annotation encoding="application/x-tex">
maxZ = 2x_1 + 3x_2 + 3.24x_3 + 0.5x_4 + 1.2x_5 + 2x_6
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="base displaystyle textstyle uncramped"><span class="mord mathit">m</span><span class="mord mathit">a</span><span class="mord mathit">x</span><span class="mord mathit" style="margin-right:0.07153em;">Z</span><span class="mrel">=</span><span class="mord mathrm">2</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin">+</span><span class="mord mathrm">3</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin">+</span><span class="mord mathrm">3</span><span class="mord mathrm">.</span><span class="mord mathrm">2</span><span class="mord mathrm">4</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin">+</span><span class="mord mathrm">0</span><span class="mord mathrm">.</span><span class="mord mathrm">5</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">4</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin">+</span><span class="mord mathrm">1</span><span class="mord mathrm">.</span><span class="mord mathrm">2</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">5</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin">+</span><span class="mord mathrm">2</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">6</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span></span></span></span></span> </p>
<center>
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><mo fence="true">{</mo><mtable><mtr><mtd><mrow><mn>0</mn><mi mathvariant="normal">.</mi><mn>1</mn><msub><mi>x</mi><mrow><mn>1</mn><mn>1</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><mn>7</mn><mi mathvariant="normal">.</mi><mn>2</mn><mn>6</mn><msub><mi>x</mi><mrow><mn>1</mn><mn>2</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><mn>0</mn><mi mathvariant="normal">.</mi><mn>2</mn><msub><mi>x</mi><mrow><mn>1</mn><mn>3</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><mn>2</mn><mi mathvariant="normal">.</mi><mn>9</mn><msub><mi>x</mi><mrow><mn>1</mn><mn>4</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><mn>4</mn><mi mathvariant="normal">.</mi><mn>4</mn><msub><mi>x</mi><mrow><mn>1</mn><mn>4</mn></mrow></msub><mo>+</mo><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><mn>3</mn><mi mathvariant="normal">.</mi><mn>3</mn><msub><mi>x</mi><mrow><mn>1</mn><mn>4</mn></mrow></msub><mtext>&#xA0;</mtext><mo>&gt;</mo><mo>=</mo><mtext>&#xA0;</mtext><mn>1</mn><mn>2</mn><mi mathvariant="normal">.</mi><mn>3</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn><mi mathvariant="normal">.</mi><mn>2</mn><mn>4</mn><msub><mi>x</mi><mrow><mn>2</mn><mn>1</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><mn>0</mn><mi mathvariant="normal">.</mi><mn>4</mn><msub><mi>x</mi><mrow><mn>2</mn><mn>2</mn></mrow></msub><mo>+</mo><mtext>&#xA0;</mtext><mn>4</mn><mi mathvariant="normal">.</mi><mn>1</mn><msub><mi>x</mi><mrow><mn>2</mn><mn>3</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><mn>1</mn><mi mathvariant="normal">.</mi><mn>1</mn><msub><mi>x</mi><mrow><mn>2</mn><mn>4</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><mn>1</mn><mn>1</mn><mi mathvariant="normal">.</mi><mn>3</mn><mn>1</mn><msub><mi>x</mi><mrow><mn>2</mn><mn>5</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><mn>0</mn><mi mathvariant="normal">.</mi><mn>1</mn><msub><mi>x</mi><mrow><mn>2</mn><mn>6</mn></mrow></msub><mtext>&#xA0;</mtext><mo>&gt;</mo><mo>=</mo><mtext>&#xA0;</mtext><mn>1</mn><mn>4</mn><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>1</mn><mi mathvariant="normal">.</mi><mn>6</mn><mn>8</mn><msub><mi>x</mi><mrow><mn>3</mn><mn>1</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><mn>0</mn><mi mathvariant="normal">.</mi><mn>1</mn><msub><mi>x</mi><mrow><mn>3</mn><mn>2</mn></mrow></msub><mo>+</mo><mtext>&#xA0;</mtext><mn>0</mn><mi mathvariant="normal">.</mi><mn>8</mn><msub><mi>x</mi><mrow><mn>3</mn><mn>3</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><mn>4</mn><mi mathvariant="normal">.</mi><mn>1</mn><msub><mi>x</mi><mrow><mn>3</mn><mn>4</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><mn>0</mn><mi mathvariant="normal">.</mi><mn>1</mn><msub><mi>x</mi><mrow><mn>3</mn><mn>5</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><mn>0</mn><mi mathvariant="normal">.</mi><mn>9</mn><msub><mi>x</mi><mrow><mn>3</mn><mn>6</mn></mrow></msub><mtext>&#xA0;</mtext><mo>&lt;</mo><mo>=</mo><mtext>&#xA0;</mtext><mn>1</mn><mn>4</mn><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn><mi mathvariant="normal">.</mi><mn>6</mn><mn>8</mn><msub><mi>x</mi><mrow><mn>4</mn><mn>1</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><mn>0</mn><mi mathvariant="normal">.</mi><mn>2</mn><msub><mi>x</mi><mrow><mn>4</mn><mn>2</mn></mrow></msub><mo>+</mo><mtext>&#xA0;</mtext><mn>0</mn><mi mathvariant="normal">.</mi><mn>1</mn><msub><mi>x</mi><mrow><mn>4</mn><mn>3</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><mn>1</mn><mn>0</mn><mi mathvariant="normal">.</mi><mn>1</mn><msub><mi>x</mi><mrow><mn>4</mn><mn>4</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><mn>5</mn><mi mathvariant="normal">.</mi><mn>9</mn><msub><mi>x</mi><mrow><mn>4</mn><mn>5</mn></mrow></msub><mtext>&#xA0;</mtext><mo>+</mo><mtext>&#xA0;</mtext><mn>1</mn><mi mathvariant="normal">.</mi><mn>1</mn><msub><mi>x</mi><mrow><mn>4</mn><mn>6</mn></mrow></msub><mtext>&#xA0;</mtext><mo>&lt;</mo><mo>=</mo><mtext>&#xA0;</mtext><mn>1</mn><mn>5</mn><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>&gt;</mo><mo>=</mo><mn>2</mn><mi mathvariant="normal">.</mi><mn>6</mn><mo separator="true">,</mo><msub><mi>x</mi><mn>3</mn></msub><mo>&lt;</mo><mo>=</mo><mn>4</mn><mn>4</mn><mi mathvariant="normal">.</mi><mn>8</mn><mo separator="true">,</mo><mtext>&#xA0;</mtext><msub><mi>x</mi><mn>4</mn></msub><mo>&gt;</mo><mo>=</mo><mn>1</mn></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \left\{
\begin{aligned}
0.1x_{11}\ +\  7.26x_{12}\ +\ 0.2x_{13}\ +\ 2.9x_{14}\ +\ 4.4x_{14} +\ +\ 3.3x_{14} \ &gt;= \ 12.3 \\
0.24x_{21}\ +\ 0.4x_{22} +\ 4.1x_{23}\ +\ 1.1x_{24}\ +\ 11.31x_{25}\  +\ 0.1x_{26}\ &gt;=\ 140 \\
1.68x_{31}\ +\ 0.1x_{32} +\ 0.8x_{33}\ +\ 4.1x_{34}\ +\ 0.1x_{35}\  +\ 0.9x_{36}\ &lt;=\ 140 \\
0.68x_{41}\ +\ 0.2x_{42} +\ 0.1x_{43}\ +\ 10.1x_{44}\ +\ 5.9x_{45}\  +\ 1.1x_{46}\ &lt;=\ 150 \\
\\
x_1 &gt;= 2.6, x_3 &lt;= 44.8,\ x_4 &gt;= 1
\end{aligned}
\right.
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:3.8500199999999998em;"></span><span class="strut bottom" style="height:7.2000399999999996em;vertical-align:-3.3500199999999998em;"></span><span class="base textstyle uncramped"><span class="minner textstyle uncramped"><span class="mopen style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing mult"><span class="vlist"><span style="top:2.45001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23A9;</span></span></span><span style="top:2.45001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:2.15001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:1.85001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:1.5500099999999999em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:1.2500099999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:0.9500099999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:-0.000010000000000287557em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23A8;</span></span></span><span style="top:-1.1500100000000002em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:-1.4500100000000002em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:-1.7500100000000003em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:-2.0500100000000003em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:-2.35001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:-2.65001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23AA;</span></span></span><span style="top:-2.95002em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="delimsizinginner delim-size4"><span>&#x23A7;</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist"><span style="top:-3.0100000000000007em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="mord textstyle uncramped"><span class="mord mathrm">0</span><span class="mord mathrm">.</span><span class="mord mathrm">1</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span><span class="mord mathrm mtight">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">7</span></span><span class="mord mathrm">.</span><span class="mord mathrm">2</span><span class="mord mathrm">6</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span><span class="mord mathrm mtight">2</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">0</span></span><span class="mord mathrm">.</span><span class="mord mathrm">2</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span><span class="mord mathrm mtight">3</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">2</span></span><span class="mord mathrm">.</span><span class="mord mathrm">9</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span><span class="mord mathrm mtight">4</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">4</span></span><span class="mord mathrm">.</span><span class="mord mathrm">4</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span><span class="mord mathrm mtight">4</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin">+</span><span class="mord"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">3</span></span><span class="mord mathrm">.</span><span class="mord mathrm">3</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span><span class="mord mathrm mtight">4</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mrel"><span class="mspace">&#xA0;</span><span class="mrel">&gt;</span></span><span class="mrel">=</span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">1</span></span><span class="mord mathrm">2</span><span class="mord mathrm">.</span><span class="mord mathrm">3</span></span></span><span style="top:-1.8100000000000007em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="mord textstyle uncramped"><span class="mord mathrm">0</span><span class="mord mathrm">.</span><span class="mord mathrm">2</span><span class="mord mathrm">4</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span><span class="mord mathrm mtight">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">0</span></span><span class="mord mathrm">.</span><span class="mord mathrm">4</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span><span class="mord mathrm mtight">2</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin">+</span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">4</span></span><span class="mord mathrm">.</span><span class="mord mathrm">1</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span><span class="mord mathrm mtight">3</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">1</span></span><span class="mord mathrm">.</span><span class="mord mathrm">1</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span><span class="mord mathrm mtight">4</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">1</span></span><span class="mord mathrm">1</span><span class="mord mathrm">.</span><span class="mord mathrm">3</span><span class="mord mathrm">1</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span><span class="mord mathrm mtight">5</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">0</span></span><span class="mord mathrm">.</span><span class="mord mathrm">1</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span><span class="mord mathrm mtight">6</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mrel"><span class="mspace">&#xA0;</span><span class="mrel">&gt;</span></span><span class="mrel">=</span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">1</span></span><span class="mord mathrm">4</span><span class="mord mathrm">0</span></span></span><span style="top:-0.6100000000000005em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="mord textstyle uncramped"><span class="mord mathrm">1</span><span class="mord mathrm">.</span><span class="mord mathrm">6</span><span class="mord mathrm">8</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">3</span><span class="mord mathrm mtight">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">0</span></span><span class="mord mathrm">.</span><span class="mord mathrm">1</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">3</span><span class="mord mathrm mtight">2</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin">+</span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">0</span></span><span class="mord mathrm">.</span><span class="mord mathrm">8</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">3</span><span class="mord mathrm mtight">3</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">4</span></span><span class="mord mathrm">.</span><span class="mord mathrm">1</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">3</span><span class="mord mathrm mtight">4</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">0</span></span><span class="mord mathrm">.</span><span class="mord mathrm">1</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">3</span><span class="mord mathrm mtight">5</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">0</span></span><span class="mord mathrm">.</span><span class="mord mathrm">9</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">3</span><span class="mord mathrm mtight">6</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mrel"><span class="mspace">&#xA0;</span><span class="mrel">&lt;</span></span><span class="mrel">=</span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">1</span></span><span class="mord mathrm">4</span><span class="mord mathrm">0</span></span></span><span style="top:0.5899999999999996em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="mord textstyle uncramped"><span class="mord mathrm">0</span><span class="mord mathrm">.</span><span class="mord mathrm">6</span><span class="mord mathrm">8</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">4</span><span class="mord mathrm mtight">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">0</span></span><span class="mord mathrm">.</span><span class="mord mathrm">2</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">4</span><span class="mord mathrm mtight">2</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin">+</span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">0</span></span><span class="mord mathrm">.</span><span class="mord mathrm">1</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">4</span><span class="mord mathrm mtight">3</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">1</span></span><span class="mord mathrm">0</span><span class="mord mathrm">.</span><span class="mord mathrm">1</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">4</span><span class="mord mathrm mtight">4</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">5</span></span><span class="mord mathrm">.</span><span class="mord mathrm">9</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">4</span><span class="mord mathrm mtight">5</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mbin"><span class="mspace">&#xA0;</span><span class="mbin">+</span></span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">1</span></span><span class="mord mathrm">.</span><span class="mord mathrm">1</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">4</span><span class="mord mathrm mtight">6</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mrel"><span class="mspace">&#xA0;</span><span class="mrel">&lt;</span></span><span class="mrel">=</span><span class="mord mathrm"><span class="mspace">&#xA0;</span><span class="mord mathrm">1</span></span><span class="mord mathrm">5</span><span class="mord mathrm">0</span></span></span><span style="top:1.7899999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="mord textstyle uncramped"></span></span><span style="top:2.99em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="mord textstyle uncramped"><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mrel">&gt;</span><span class="mrel">=</span><span class="mord mathrm">2</span><span class="mord mathrm">.</span><span class="mord mathrm">6</span><span class="mpunct">,</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mrel">&lt;</span><span class="mrel">=</span><span class="mord mathrm">4</span><span class="mord mathrm">4</span><span class="mord mathrm">.</span><span class="mord mathrm">8</span><span class="mpunct">,</span><span class="mord"><span class="mspace">&#xA0;</span><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">4</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span><span class="mrel">&gt;</span><span class="mrel">=</span><span class="mord mathrm">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">&#x200B;</span></span>&#x200B;</span></span></span></span></span><span class="mclose sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span>
</center>

<h4 id="r&#x6C42;&#x89E3;&#x8FC7;&#x7A0B;&#x5982;&#x4E0B;">R&#x6C42;&#x89E3;&#x8FC7;&#x7A0B;&#x5982;&#x4E0B;:</h4>
<pre><code class="lang-r"><span class="hljs-keyword">library</span>(<span class="hljs-string">&apos;lpSolveAPI&apos;</span>, character.only=<span class="hljs-literal">TRUE</span>)

lprec &lt;- make.lp(<span class="hljs-number">0</span>, <span class="hljs-number">6</span>)
set.objfn(lprec, c(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">3.24</span>, <span class="hljs-number">0.5</span>, <span class="hljs-number">1.2</span>, <span class="hljs-number">2</span>))

add.constraint(lprec, c(<span class="hljs-number">0.1</span>, <span class="hljs-number">7.26</span>, <span class="hljs-number">0.2</span>, <span class="hljs-number">2.9</span>, <span class="hljs-number">4.4</span>, <span class="hljs-number">3.3</span>), <span class="hljs-string">&quot;&gt;=&quot;</span>, <span class="hljs-number">12.3</span>)
add.constraint(lprec, c(<span class="hljs-number">0.24</span>, <span class="hljs-number">4.1</span>, <span class="hljs-number">0.4</span>, <span class="hljs-number">1.1</span>, <span class="hljs-number">11.31</span>, <span class="hljs-number">0.1</span>), <span class="hljs-string">&quot;&gt;=&quot;</span>, <span class="hljs-number">140</span>)
add.constraint(lprec, c(<span class="hljs-number">1.68</span>, <span class="hljs-number">0.1</span>, <span class="hljs-number">0.8</span>, <span class="hljs-number">4.1</span>, <span class="hljs-number">0.1</span>, <span class="hljs-number">0.9</span>), <span class="hljs-string">&quot;&lt;=&quot;</span>, <span class="hljs-number">140</span>)
add.constraint(lprec, c(<span class="hljs-number">0.68</span>, <span class="hljs-number">0.2</span>, <span class="hljs-number">0.1</span>, <span class="hljs-number">10.1</span>, <span class="hljs-number">5.9</span>, <span class="hljs-number">1.1</span>), <span class="hljs-string">&quot;&lt;=&quot;</span>, <span class="hljs-number">150</span>)

set.bounds(lprec, lower = c(<span class="hljs-number">2.6</span>, <span class="hljs-number">1</span>), columns = c(<span class="hljs-number">1</span>, <span class="hljs-number">4</span>))
set.bounds(lprec, upper = <span class="hljs-number">44.8</span>, columns = <span class="hljs-number">3</span>)

RowNames &lt;- c(<span class="hljs-string">&quot;ROW_1&quot;</span>, <span class="hljs-string">&quot;ROW_2&quot;</span>, <span class="hljs-string">&quot;ROW_3&quot;</span>,<span class="hljs-string">&quot;ROW_4&quot;</span>)
ColNames &lt;- c(<span class="hljs-string">&quot;COL_1&quot;</span>, <span class="hljs-string">&quot;COL_2&quot;</span>, <span class="hljs-string">&quot;COL_3&quot;</span>, <span class="hljs-string">&quot;COL_4&quot;</span>,<span class="hljs-string">&quot;COL_5&quot;</span>,<span class="hljs-string">&quot;COL_6&quot;</span>)

dimnames(lprec) &lt;- list(RowNames, ColNames)

solve(lprec)

get.objective(lprec)

get.variables(lprec)

get.constraints(lprec)

get.solutioncount(lprec)
</code></pre>
<p>&#x7EC8;&#x7AEF;&#x6253;&#x5370;:</p>
<pre><code class="lang-r">master:optimization_se dreamboad$ Rscript linear_programming.R
[<span class="hljs-number">1</span>] <span class="hljs-number">0</span>
[<span class="hljs-number">1</span>] <span class="hljs-number">20.37119</span>
[<span class="hljs-number">1</span>]  <span class="hljs-number">2.60000</span>  <span class="hljs-number">0.00000</span>  <span class="hljs-number">0.00000</span>  <span class="hljs-number">1.00000</span> <span class="hljs-number">12.22599</span>  <span class="hljs-number">0.00000</span>
[<span class="hljs-number">1</span>]  <span class="hljs-number">56.954377</span> <span class="hljs-number">140.000000</span>   <span class="hljs-number">9.690599</span>  <span class="hljs-number">84.001369</span>
[<span class="hljs-number">1</span>] <span class="hljs-number">1</span>
</code></pre>
<hr>
<p>&#xA0;&#xA0;&#xA0;&#xA0; <strong>&#x603B;&#x7ED3;&#x3001;&#x8BF4;&#x660E;</strong></p>
<pre><code>1&#x3001; &#x4ECE;&#x4E0A;&#x4F8B;&#x53EF;&#x4EE5;&#x770B;&#x5230;lpSolve&#x53EF;&#x4EE5;&#x8F7B;&#x677E;&#x6C42;&#x89E3;&#x7EBF;&#x6027;&#x89C4;&#x5212;&#x95EE;&#x9898;&#xFF0C;&#x6700;&#x4F18;&#x89E3;&#x662F;&#x901A;&#x8FC7;&#x5355;&#x7EAF;&#x578B;&#x3001;&#x5206;&#x652F;&#x5B9A;&#x754C;&#x7B97;&#x6CD5;&#x7B49;&#x6765;&#x5B9E;&#x73B0;
2&#x3001; &#x73B0;&#x5B9E;&#x751F;&#x6D3B;&#x4E2D;&#xFF0C;&#x5EFA;&#x6A21;&#x591A;&#x53D7;&#x4E00;&#x4E9B;&#x51B3;&#x7B56;&#x53D8;&#x91CF;&#x5FC5;&#x987B;&#x4E3A;&#x6574;&#x578B;&#x3001;&#x4E00;&#x4E00;&#x5BF9;&#x5E94;&#x7684;&#x7EA6;&#x675F;&#xFF0C;&#x5982;&#x6307;&#x6D3E;&#x95EE;&#x9898;&#x3001;&#x8FD0;&#x8F93;&#x95EE;&#x9898;&#xFF0C;&#x6B64;&#x65F6;&#x5219;&#x9700;&#x8981;&#x5BF9;&#x6A21;&#x578B;
  &#x505A;&#x4E00;&#x4E9B;&#x9488;&#x5BF9;&#x6027;&#x7684;&#x6539;&#x8FDB;
3&#x3001; &#x7EBF;&#x6027;&#x89C4;&#x5212;&#x4E2D;&#xFF0C;&#x7AD9;&#x5728;&#x4E24;&#x4E2A;&#x4E0D;&#x540C;&#x7684;&#x7ACB;&#x573A;&#xFF0C;&#x5F15;&#x53D1;&#x539F;&#x95EE;&#x9898;&#x4E0E;&#x5BF9;&#x5076;&#x95EE;&#x9898;&#x7684;&#x601D;&#x8003;&#xFF0C;&#x4F46;&#x4E24;&#x8005;&#x7684;&#x89E3;&#x7A7A;&#x95F4;&#x5173;&#x7CFB;&#x662F;&#x4E00;&#x81F4;&#x7684;&#xFF0C;&#x901A;&#x8FC7;&#x7AD9;&#x5728;&#x5BF9;&#x5076;
  &#x95EE;&#x9898;&#x7684;&#x5F71;&#x5B50;&#x4EF7;&#x683C;&#x3001;&#x7075;&#x654F;&#x5EA6;&#x7B49;&#x66F4;&#x9AD8;&#x7EF4;&#x5EA6;&#x6765;&#x4E0A;&#x6765;&#x770B;&#x5F85;&#x8D44;&#x6E90;&#x7EA6;&#x675F;&#xFF0C;&#x6316;&#x6398;&#x5176;&#x4EF7;&#x503C;&#x5185;&#x6DB5;&#x3002;
</code></pre><h3 id="&#x4E09;&#x3001;&#x53C2;&#x8003;">&#x4E09;&#x3001;&#x53C2;&#x8003;</h3>
<ul>
<li><a href="https://cran.r-project.org/web/packages/lpSolve/lpSolve.pdf" target="_blank">1&#x3001;R lpSolve&#x6587;&#x6863;</a></li>
<li><a href="https://cran.r-project.org/web/packages/lpSolveAPI/lpSolveAPI.pdf" target="_blank">2&#x3001;R lpSolve api&#x6587;&#x6863;</a> </li>
</ul>
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